Reachability in K 3,3 -free and K 5 -free Graphs is in Unambiguous Logspace

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1 CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages Reachability in K 3,3 -free and K 5 -free Graphs is in Unambiguous Logspace Thomas Thierauf Fabian Wagner Received May 30, 2012; Revised October 14, 2013, February 8, 2014, March 3, 2014, received in final form April 10, 2014; Published April 14, 2014 Abstract: We show that the reachability problem for directed graphs that are either K 3,3 -free or K 5 -free is in unambiguous log-space, UL coul. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL coul. Our algorithm decomposes the graphs into biconnected and triconnected components. This gives a tree structure on these components. The non-planar components are replaced by planar components that maintain the reachability properties. For K 5 -free graphs we also need a decomposition into 4-connected components. Thereby we provide a logspace reduction to the planar reachability problem. We show the same upper bound for computing distances in K 3,3 -free and K 5 -free directed graphs and for computing longest paths in K 3,3 -free and K 5 -free directed acyclic graphs. Key words and phrases: Reachability, logspace, planar graphs, K 3,3 -free graphs, K 5 -free graphs 1 Introduction In this paper we consider the reachability problem on graphs: Reachability Input: a graph G = (V,E) and two vertices s,t V. Question: is there a path from s to t in G? Supported by DFG grants TH 472/4-1 and TO 200/ Thomas Thierauf and Fabian Wagner cb Licensed under a Creative Commons Attribution License (CC-BY) DOI: /cjtcs

2 THOMAS THIERAUF AND FABIAN WAGNER The complexity of Reachability is essentially solved for directed and undirected graphs. In general, i.e. for directed graphs, Reachability is complete for the class nondeterministic logspace, NL. For undirected graphs, the complexity was open for a long time, until Reingold [22] showed in a break-through result that Reachability is complete for the class logspace, L. This result is one of the major tools in our paper. The complexity of Reachability in planar graphs is still not completely settled. Bourke, Tewari and Vinodchandran [7] proved that Reachability on planar graphs is in the class unambiguous nondeterministic logspace, UL coul. It is also known to be hard for L. They built on work of Reinhard and Allender [23] and Allender, Datta, and Roy [2]. Jacoby and Tantau [16] showed that for series-parallel graphs, reachability is complete for L. More recently, Allender et.al. [1] showed that Reachability for graphs embedded on the torus is logspace reducible to the planar case. Kynčl and Vyskočil [18] generalized this result to graphs embedded on a fixed surface of arbitrary genus. Das, Datta and Nimbhorkar [9] showed that Reachability for graphs with bounded treewidth is in logspace, when a tree decomposition is given. Elberfeld, Jacoby and Tantau show in [12] that a tree decomposition can be computed in logspace. We study Reachability on extensions of planar graphs. Our main result is logspace reduction from Reachability for K 3,3 -free and K 5 -free graphs to planar Reachability. Thus, the current upper bound for planar Reachability, UL coul, carries over to Reachability for K 3,3 -free and K 5 -free graphs. One motivation for our results is to improve the complexity upper bounds of certain reachability problems, from NL to UL in this case. The major open question is whether one can extend our results further such that we finally get a collapse of NL to UL. Our technique is to decompose a given graph G into its triconnected components. We show that this can be done in logspace, even for general graphs. Then we exploit the properties of the triconnected components. In the case of a K 3,3 -free graph G, Asano [4] showed that the triconnected components of G are either planar or the K 5. In the case of a K 5 -free graph G, it follows from a theorem of Wagner [26] (cf. Khuller [17]) that there can be nonplanar triconnected components of G of two types only: either they are isomorphic to the Möbius ladder M 8, (see Figure 6 on page 17), or they can be decomposed into 4-connected components which are all planar. The M 8 contains a K 3,3 and is therefore nonplanar. We also show that the decomposition into 4-connected components in the last item can be done in logspace. Because we want to reduce the reachability problem to planar reachability, the obstacles we have at this point are the K 5 -components in the K 3,3 -free case and the M 8 -components in the K 5 -free case. We construct planar gadget graphs which we use to replace these nonplanar components. Then we recombine the components. There are several restrictions for the gadgets that have to be taken care of. Clearly, the original reachability problem should not be altered by the replacement. To make the gadgets planar, some edges of the original graphs occur several times as copies. With the copied edges, we also copy subgraphs CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

3 REACHABILITY IN K 3,3 -FREE AND K 5 -FREE GRAPHS IN UL attached to the endpoints of the edges. Since the replacement is done recursively in the construction algorithm, the gadget has to be designed in a way that large subgraphs are not copied. Otherwise this would not work in logspace. We also consider the problem of computing distances in a graph. Jacoby and Tantau [16] showed that for series-parallel graphs the distance problem is complete for L. Thierauf and Wagner [24] proved that the distance problem for planar graphs is in UL coul. We will see that our transformations from K 3,3 -free or K 5 -free graphs to planar graphs maintain not just reachability, but also the distances between vertices. Therefore it follows from our results that distances in K 3,3 -free or K 5 -free graphs can be computed in UL coul. Another related problem is to compute longest paths. In general, it is NP-complete, for directed and undirected graphs. But it is NL-complete for directed acyclic graphs (DAG). For series-parallel graphs it is complete for L [16]. Limaye, Mahajan and Nimbhorkar [19] prove that longest paths in planar DAGs can be computed in UL coul. Our transformations from K 3,3 -free or K 5 -free graphs to planar graphs also maintain longest paths between vertices. This is easy to see in the case of K 3,3 -free DAGs and requires some extra arguments in the case of K 5 -free DAGs. Hence, longest paths in K 3,3 -free or K 5 -free DAGs can be computed in UL coul. The paper is organized as follows. Section 2 provides definitions and notations. In Section 3 we show how to decompose a graph into its biconnected and trionnected components in logspace. In Section 4 and 5 we prove that reachability on K 3,3 -free and K 5 -free graphs reduces to reachability on planar graphs, respectively. In these sections we also show the results on distances and longest paths. 2 Definitions and Notations Complexity classes. The class L is the class of languages accepted by deterministic logspace Turing machines and NL by nondeterministic logspace Turing machines. The class UL contains languages accepted by unambiguous nondeterministic logspace machines, i.e. there exists at most one accepting computation path. Whereas NL is known to be closed under complement, this is not known for UL. Therefore we define coul as the class of complements of languages in UL. The class L is closed under Turing reductions, i.e. L L = L. Similarly L UL coul = UL coul. The composition of two logspace computable functions is computable in logspace. By L m and L T we denote logspace many-one and Turing reduction, respectively. Graphs. A graph G = (V,E) consists of a finite set of vertices V (G) = V and edges E(G) = E V V. For U V let G U be the induced subgraph of G on V U. A graph G is called undirected if E is symmetric. An undirected graph G is connected if there is a path between any two vertices in G. Let G be undirected and S V with S = k. We call S a k-separating set, if G S is not connected. For u,v V we say that S separates u from v in G, if u S, v S, or u and v are in different components of G S. For sets of vertices V 1,V 2 V we say that S separates V 1 from V 2 in G, if S separates every v 1 V 1 from every v 2 V 2. A k-separating set is called articulation point (or cut vertex) for k = 1, separating pair for k = 2, and separating triple for k = 3. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

4 THOMAS THIERAUF AND FABIAN WAGNER A graph G is k-connected if it contains no (k 1)-separating set. Hence a 1-connected graph is simply a connected graph. A 2-connected graph is also called biconnected, a 3-connected graph is also called triconnected. Let S be a k-separating set in a k-connected graph G. Let G be a connected component in G S. A split graph or a split component of S in G is the induced subgraph of G on vertices V (G ) S, where we add virtual edges between all pairs of vertices in S. Note that the vertices of a separating set S can occur in several split graphs of G. A K 3,3 -free graph is an undirected graph which does not contain a K 3,3 as a minor. A K 5 -free graph is an undirected graph which does not contain a K 5 as a minor. In particular, planar graphs are K 3,3 -free and K 5 -free [26]. Reachability problems. to G. Let G be a class of graphs. We consider the following problems restricted G-Reachability = {(G, s, t) G G contains a path from s to t } G-Distance = {(G,s,t,k) G G contains a path from s to t of length k } G-Long-Path = {(G,s,t,k) G G contains a simple path from s to t of length k } As already mentioned in the introduction, the following results are known. Reachability is NL-complete, undirected Reachability is L-complete [22], planar Reachability is in UL coul [7]. By the second item one can find out whether an undirected graph G is connected: cycle through all pairs of vertices of the graph and check reachability for each pair. Therefore graph connectivity is in L. Recall that a set S of vertices is a separating set in G if G S is not connected. Hence we can check in logspace whether S is a separating set. For constant k, a logspace machine can cycle through all size k subsets of vertices and output the k-separating ones. Hence, in particular, all articulation points, separating pairs, and separating triples of a graph can be computed in logspace. This is what we will use later on. 3 Decomposition of a Graph into Component Trees In this section we show how to split a graph G into connected, biconnected and triconnected components. Within this context, we always refer to the undirected version of G. That is, every edge in G is considered as an undirected edge, and we still call it G. Since the reachability problem on undirected graphs is in logspace [22], we can check whether s and t are in the same connected component of G, and ignore all other components. Therefore we may w.l.o.g. assume that G is connected. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

5 REACHABILITY IN K 3,3 -FREE AND K 5 -FREE GRAPHS IN UL We further decompose G into biconnected and triconnected components. There is an extensive literature on graph decomposition, see for example [25, 14, 15, 5, 6, 21]. We give definitions that are adapted to a logspace computation of the decompositions. As we already mentioned, logspace computable functions are closed under composition. Hence we may separately argue that the decomposition into biconnected components is in logspace and the decomposition of biconnected components into triconnected components is in logspace. Then it follows that also the whole process is in logspace. 3.1 The biconnected component tree We decompose G into biconnected components by splitting G at all its articulation points. Definition 3.1. Let G = (V,E) be a connected graph. A biconnected component of G is a is a maximal biconnected subgraph of G. Observe that an articulation point occurs in 2 components. The intersection of the vertices of two biconnected components is either empty or an articulation point. Lemma 3.2. The biconnected components of a connected graph G can be computed in logspace. Proof. As explained at the end of Section 2, we can find out in logspace whether two vertices u,v of G belong to the same biconnected component. Hence, for a given vertex v, we can compute in logspace all vertices u of G which are in the same biconnected component as v. We use this as a subroutine in our algorithm which outputs all biconnected components of G. The main loop of the algorithm cycles over all vertices of G. Let v be the current vertex. If v is the first vertex in the loop, we compute all vertices that are in the same biconnected component as v. For all other v s we check whether v is in a biconnected component of some vertex u < v. In this case we have already output v in an earlier stage and proceed to the next v in the loop. Otherwise v is in a new component and we compute the vertices which are in the same biconnected component as v. We define a graph with the biconnected components as nodes. Definition 3.3. The biconnected component tree of G is the following graph. There is a node for every biconnected component and for every articulation point of G. There is an edge between the node for biconnected component B and the node for an articulation point a, if a belongs to B. As a convention in this paper, we use the term node for the nodes of the component trees and vertex for the vertices of the input graph G and its transformations. In a slight abuse of notation, we also denote the node for component B in the tree by the same name, B, instead of for example v B, and similar for articulation points. It should always be clear from the context what is meant. Note that the biconnected component tree is in fact a tree: it is connected because G is connected, and it is acyclic because we deal with articulation points. We define an order on the nodes of the biconnected component tree of G: The logspace algorithms that compute the articulation points and the biconnected components of G make their respective outputs CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

6 THOMAS THIERAUF AND FABIAN WAGNER in a certain order. We use this order for the tree nodes. I.e., as the root of the tree we choose the first articulation point. The order on the children of a node are defined by the order they appear in the construction algorithms. We will use this order when we navigate in the component tree. A trivial case is when G has no articulation points. Then G is biconnected and the component tree consists of just one node. In this case we can directly proceed to Section 3.2. Hence, for the rest of this section we assume that there are articulation points in G. By Lemma 3.2 we can compute the nodes of the component tree in logspace. We show that we can also traverse the tree in logspace. It is known that trees can be traversed in logspace, see [8, 20]. We show how to do this in case of component trees. Lemma 3.4. The biconnected component tree of a connected graph G can be traversed in logspace. Proof. The traversal proceeds as a depth-first search. We show how to navigate locally in the component tree, i.e., for a current node how to compute its parent, first child, and next sibling. We explore the tree starting at the root. Thereby we store the following information on the tape. We always store the root node, i.e., one vertex which is an articulation point. When the current node is articulation point a 0, we just store it. When the current node is a biconnected component B with parent articulation point a 0, then we store a 0 and an arbitrary vertex v a 0 from B. For the last item, note that we cannot afford to store all vertices of B. The vertex v that we store serves as a representative for B. As a choice for v take the first vertex of B that is computed by the construction algorithm of Lemma 3.2. Note that v and a 0 together with the root node identify B uniquely. The traversal continues by exploring the subtrees at the articulations point in B, different from a 0. Let a 1 be the current articulation point in B. We compute a representative vertex for the first biconnected split component of a 1 different from B. Then we erase a 0 and the representative vertex for B from the tape and recursively traverse the subtrees at a 1. When we return from the subtrees at a 1, we recompute a 0 and B, the parent of a 1. This is done by computing the path from the root node to B in the component tree. That is, we start at the root node and look for the child component that contains B via reachability queries. Then we iterate the search until we reach B, where we always store the current parent node. The tree traversal continues with the next sibling of B in the tree. That is, we compute the next articulation point in B after a 1 with respect to the order on the articulation points. Then we delete a 1 from the work tape. If B does not have a next sibling, we return to the parent of B. Lemma 3.4 allows us to reduce the the reachability problem for connected graphs to the one for biconnected graphs. Lemma 3.5. Reachability L T biconnected Reachability. The reduction maintains planarity, K 3,3-freeness, and K 5 -freeness. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

7 REACHABILITY IN K 3,3 -FREE AND K 5 -FREE GRAPHS IN UL Proof. Let B s be a biconnected component of G that contains s. If s itself is an articulation point, there might be several components that contain s. In this case choose any such component. Let similarly B t be a component for t. There is a unique simple path P from B s to B t in the biconnected component tree. Such a path can be computed in logspace because this is a reachability problem in an undirected graph, the biconnected component tree, which we can traverse by Lemma 3.4. Let a 1,a 2,...,a k be the articulation points where P passes through, in this order. The crucial observation now is, that a simple path p from s to t in G has to go through the articulation points a 1,a 2,...,a k in the same order. Moreover, the part of p between a i and a i+1 stays within the biconnected component defined by a i and a i+1 : suppose that the path would deviate on the way and go through some other articulation point a to a neighboring component. But then p would have to go through a again and hence, p would not be simple. Therefore there is a path from s to t in G if, and only if, there are paths from a i to a i+1 in the biconnected component between a i and a i+1, for i = 1,...,k 1, and similarly, between s and a 1 and between a k and t. As a consequence of the lemma, it suffices to consider biconnected graphs in the following. 3.2 The triconnected component tree We further decompose the biconnected graph G into its triconnected components. In contrast to the decomposition presented in earlier versions of this paper, we found an easier way to argue later on, together with Datta and Nimbhorkar [11]. For completeness, and also because we need the structure of the decomposition later on, we present the new approach from [11] here. An obvious approach to decompose a biconnected graph G into 3-connected components would be to split G at every separating pair. However, there are some subtleties to take care of. Consider for example a simple cycle. Every pair of vertices in the cycle, except the neighboring ones, constitute a separating pair. Hence, if we would split the cycle at every separating pair, pieces of the cycle would be in many components. To avoid this, we look for such cycle components and do not split them any further. The vertices of separating pair {a,b} lie on a cycle if there are only two vertex disjoint paths between them. We decompose a biconnected graph only along separating pairs which are connected by at least three disjoint paths. Definition 3.6. [11] Let G = (V,E) be a biconnected graph. A separating pair {a,b} is called 3-connected if there are three vertex-disjoint paths between a and b in G. The triconnected components of G are the split graphs we obtain from G by splitting G successively along all 3-connected separating pairs, in any order. If a separating pair {a,b} is connected by an edge in G, then we also define a 3-bond for {a,b} as a triconnected component, i.e., a multigraph with two vertices {a,b} and three edges between them. In summary, we get three types of triconnected components of a biconnected graph: 3-connected components, cycle components, and 3-bonds. The task of the 3-bonds is only to represent edges of the graph which are replaced by virtual edges in the other components. That way, we do not need access to the input graph G anymore for further computation, all the information about G is available in the components. Note that the cycle components are not 3-connected. A special case is when G has separating pairs, but none of them is 3-connected. Then G is a simple cycle and constitutes one cycle component. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

8 THOMAS THIERAUF AND FABIAN WAGNER Definition 3.6 leads to the same triconnected components as defined by Hopcroft and Tarjan [15], but by a different construction: they first completely decompose the graph along all separating pairs and then merge triangles to larger cycles. They also show that the decomposition is unique, i.e., independent of the order of the separating pairs in the definition. This is also shown in [11]. Hopcroft and Tarjan [15] presented a linear-time algorithm to compute such a decomposition. Miller and Ramachandran [21] present a linear time algorithm for computing triconnected components which also has a parallel implementation on a CRCW-PRAM with O(log 2 n) parallel time and using a linear number of processors. By the next lemma, the decomposition can also be computed in logspace. Lemma 3.7. [11] The 3-connected separating pairs and the triconnected components of a biconnected graph G can be computed in logspace. Proof. We already explained that we can compute all separating pairs of G in logspace. Among those, we identify the 3-connected ones as follows. A separating pair {a,b} in G is not 3-connected if there are exactly two split components of {a,b} (without attaching virtual edges) and both are not biconnected. To see this, note that a split component C which is not biconnected has an articulation point c. All paths from a to b in C must go through c. Hence there are no two vertex disjoint paths from a to b. To check the above conditions we have to find articulation points in the split components of {a,b}. This can be done in logspace with queries to reachability. It remains to compute the vertices of a 3-connected component. Two vertices u,v V belong to the same 3-connected component or cycle component, if no 3-connected separating pair separates u from v. This property can again be checked by solving several reachability problems. In the same way as we defined the biconnected component tree of a connected graph, we define the triconnected component tree of a biconnected graph. Definition 3.8. The triconnected component tree T of G is the following graph. There is a node for each triconnected component and for each 3-connected separating pair of G. There is an edge in T between the node for triconnected component C and the node for a separating pair {a,b}, if a,b belong to C. Note that graph T is connected, because G is biconnected, and acyclic. Hence T is a tree. For an example see Figure 1. We consider an arbitrary separating pair as the root node of T. If G has no separating pair, i.e. G is 3-connected, then T consists of a single node. By Lemma 3.7 we can compute the nodes of the component tree in logspace. We show that we can also traverse the tree in logspace. Lemma 3.9. The triconnected component tree of a biconnected graph G can be computed and traversed in logspace. Proof. The traversal of the tree works analogously as in Lemma 3.4 for the biconnected component tree. Instead of articulation points we have separating pairs now. Hence we store two vertices instead of one. In case of a 3-connected component or a cycle, we again store a representative vertex from the component, for example the first vertex from the component computed by the construction algorithm. The local navigation now can be done in the same way as for the biconnected component tree. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

9 REACHABILITY IN K 3,3 -FREE AND K 5 -FREE GRAPHS IN UL a c f b e G 1 a f b a c b c e T G 1 a b G 2 d c d d d c d G G 3 G 2 G 4 G 3 G 4 Figure 1: ([10]) The decomposition of a biconnected planar graph G. Its triconnected components are G 1,...,G 4 and the corresponding decomposition into the triconnected tree T of G. The separating pairs are {a,b} and {c,d}. Since the separating pair {c,d} is connected by an edge in G, we also get {c,d} as 3-bond G 3. The virtual edges corresponding to the separating pairs are drawn with dashed lines. We define the size of a triconnected component tree in the obvious way. For our logspace algorithms it will be important to detect large children in a tree. Definition Let T be a triconnected component tree of graph G. The size of an individual component node of T is the number of vertices of G in the component. The size of T, denoted by T, is the sum of the sizes of its component nodes. Let T C be a component tree rooted at some component C and let T C be a subtree of T rooted at a child C of C. We call C a large child of C, if T C > T C /2. Clearly a node can have at most one large child. 3.3 Partitioning the reachability problem Let G = (V,E) be a biconnected graph and s,t V. Let T G be the triconnected component tree of G. Let S and T be 3-connected components that contain s and t, respectively. If s or t belong to a separating pair, they occur in several components. In this case choose an arbitrary such component. Consider the simple path from S to T in T G, say S = C 1, C 2,..., C l = T, see Figure 2. By the definition of the triconnected component tree, the nodes are alternating 3-connected component nodes and separating pair nodes. Let C i = {a i,b i } be a separating pair node, and C i 1 and C i+1 component nodes. Observe that a simple path p from s to t in G has to visit at least one vertex of each of these separating pairs. Once p has reached C i, say via a i, it will not go back to C i 1 : the only way back to C i 1 would be via b i. Then p already visited both vertices of C i and still should proceed via C i. But this is not possible because p is a simple path. Similarly, if p has reached C i+1, it will not go back to C i. Path p might pass through the components indicated below the C i s in Figure 2. But the components C 1, C 2,..., C l are visited in this order. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

10 THOMAS THIERAUF AND FABIAN WAGNER S s C 2 a 2 b 2 C 3 T t T C2 T C3 Figure 2: The triconnected component tree T G partitioned according to a path from S to T, indicated by the dashed boxes. The solid boxes indicate component nodes and the triangles indicate subtrees. Components C i are alternating separating pairs, C i = {a i,b i }, or 3-connected. For a node C i, we define the tree T Ci and its underlying graphs G i as T Ci = the subtree of T G rooted at C i, where the branches to C i 1 and C i+1 are cut off, G i = the graph corresponding to T Ci. We have just argued that the reachability problem in G can be partitioned into reachability problems in the G i s. Lemma Any simple path p from s to t in G can be written as a concatenation of paths, p = p 1 p 2 p l, such that p 1 goes from s to a 2 or b 2 in G 1, if C i is a component node, p i is a path from a i 1 or b i 1 to a i+1 or b i+1 in G i, if C i is a separating pair node, p i is a path from a i to b i in G i, or vice versa, or a trivial path p i = (a i ) or p i = (b i ), p l is a path from a l 1 or b l 1 to t in G l. In the reachability problem for G i, we search for a path from a i or b i to a i+1 or b i+1, if C i is a component node, and from a i to b i or vice versa if C i is a separating pair node. Recall that each separating pair a,b is connected by a virtual edge. Hence we might have a virtual edge (a,b) in C i on our path. If (a,b) is also a directed edge in G then we are fine. This edge is indicated by a 3-bond node as a child in the tree. Otherwise we have to check whether there is a path from a to b in G i by traversing a child of C i in T Ci. Note that in the child component the same situation may occur again. Suppose we reach vertex c of separating pair {c,d} in a child component. If the path now goes further down into a split component of {c,d}, then the only way back to b goes via d. Hence, for the separating pairs {c,d} inside the subtrees T Ci, we only want to find out whether there is a path from c to d. We do not CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

11 REACHABILITY IN K 3,3 -FREE AND K 5 -FREE GRAPHS IN UL need to consider four connectivity problems between two separating pairs as above between the root components of the T Ci s. If a component is planar, we can test reachability in UL coul [7]. However, this is not known for the nonplanar components. Allender and Mahajan [3] showed that planarity can be tested in logspace. Hence we can find out which components are planar and which are not. In the next two sections we show for K 3,3 -free and K 5 -free graphs G how these nonplanar components may look like and how to replace them by planar components such that the original reachability problem stays the same. The reason for the partitioning of the reachability problem presented above is that we do a different replacement if the nonplanar component is the root C i of T Ci than if it is inside T Ci. The difference is due to the point described in the second item above. 4 Reachability in K 3,3 -free Graphs We give a logspace reduction from the reachability problem for biconnected K 3,3 -free graphs to the reachability problem for planar graphs. The latter problem is known to be in UL coul [7]. Theorem 4.1. biconnected K 3,3 -free Reachability L m planar Reachability. We prove Theorem 4.1 in Section 4.1 and 4.2. Let G be the given biconnected K 3,3 -free graph. Consider the decomposition of the underlying undirected version of G into triconnected components as described in Section 3. Asano [4] (see also Hall [13]) showed that every nonplanar component is precisely the K 5. Figure 3 shows an example. u 3 u 4 u 5 G u 6 u 1 D T u 1 u 1 u 3 D u 6 u 2 u 3 u 6 u 4 u 5 u 3 u 6 v 1 v 5 v 3 D v 4 u 2 v2 u 2 u 1 u 2 v 1 v 2 v v 2 1 v 1 v 4 D v 2 v 4 v 1 v 1 v 3 v 4 v 5 Figure 3: The K 3,3 -free graph G contains two K 5 -components K 1 and K 2 which appear as nodes in the triconnected component tree T. To keep the figure simple, we do not show the 3-bonds and also did not further decompose the planar components. Theorem 4.2. [4] Each triconnected component of a K 3,3 -free biconnected graph is either planar or exactly the graph K 5. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

12 THOMAS THIERAUF AND FABIAN WAGNER The key step in the reduction is to replace the K 5 -components by planar components such that the reachability properties are maintained with respect to the directed graph G. 4.1 Transforming a K 5 -component into a planar component. Recall the partitioning of the reachability problem shown in Section 3.3. Let S = C 1, C 2,..., C l = T be the simple path from S to T in the triconnected component tree T of G. Let T Ci be the subtree rooted at C i and G i be the subgraph of G corresponding to T Ci. We start with the root C i and traverse the tree in depth first manner. When we reach a K 5 -component node K, then we replace it locally by a planar component such that the reachability properties do not change. As explained in Section 3.3 and Lemma 3.11, we have to solve a reachability problem in every component. In case that K is the root of T Ci, then there are actually four reachability problems. We postpone this case to Section 4.2. So let K be an inner K 5 -component node in T Ci with vertices w 1,...,w 5. There is a parent separating pair, say {w 1,w 2 }, and we search for a path from w 1 to w 2. The planar graph K that replaces K is defined as shown in Figure 4. Node K might have a large child within T Ci. Figure 4 shows K for the case when the large child is at (w 3,w 4 ), if there is one. K w 5 w 1 K w 2 w 3 w 1 w 3 w 5 w 5 w 2 w 5 (a) w 4 w 4 (b) Figure 4: (a) A K 5 -component node K. (b) The planar component node K constructed from K for the case that we search for a path from w 1 to w 2, and, if K has a large child, it is at (w 3,w 4 ). The two vertices w 5 and w 5 are copies of w 5. For example, an edge (w 1,w 5 ) in K occurs twice in K, as (w 1,w 5 ) and (w 1,w 5 ). In case (w 1,w 5 ) is a virtual edge in K, then both copies in K are considered as virtual edges. The edges of K and K are drawn undirected to not overload the picture. The edges that come from graph G have the same direction as in G. The following Lemma summarizes the important properties of K that we will use later on. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

13 REACHABILITY IN K 3,3 -FREE AND K 5 -FREE GRAPHS IN UL Lemma 4.3. Let K be a K 5 -component node with vertices w 1,...,w 5, where we search for a path from w 1 to w 2, and, if there is a large child, it is at (w 3,w 4 ). Let K be the component constructed from K shown in Figure 4. Component K has the following properties: K is planar. Every path from w 1 to w 2 in K exists as well in K, possibly going through one of the copies w 5 or w 5 instead of w 5. K contains the edge (w 3,w 4 ) only once. Vertices w 1 and w 2 are on the outer face of K in the embedding shown in Figure 4. The second item of the lemma implies the correctness of the construction: There is a path from w 1 to w 2 in K or vice versa, iff there is such a path in K. The last item will be important when we reverse the decomposition process. Then the planar components will be sticked together at the separating pairs. For the resulting graph to be planar we need w 1 and w 2 on the outer face. The price we pay for getting K planar is that one vertex of K occurs three times as copies of the original vertex, this is w 5 in Figure 4. As a consequence, some of the original edges occur now twice in K, like (w 1,w 5 ) and (w 1,w 5 ). Now we might run into a problem. Namely, at the virtual edges, our algorithm recursively explores the subtrees at this edge. If we have a copy of such an edge, we will explore the same subtree again when we come to the copy. This is fine as long as the size of the subtree is small i.e. a fraction of N, where N is the size of the subtree rooted at K in T Ci. However, there might be a large child, i.e. a child of size > N/2, see Definition 3.10 on page 9. In this case, we should not make copies of the edge in order to stay in logspace. The definition of K given in Figure 4 refers to the case that we search a path from w 1 to w 2 in K and we have a large child at (w 3,w 4 ). The same construction works if there is no large child, and it can be easily adapted to the case that the large child is at another pair, e.g. at (w 2,w 4 ). 4.2 Replacing the K 5 -components Recall that G i is the subgraph of G corresponding to component tree T Ci according to the partitioning given in Section 3.3. Our goal now is to replace the K 5 -components in T Ci by the above planar gadget and then to reassemble the components to a planar graph G i. This is done by identifying the copies of the vertices of the separating pairs. By the last item of Lemma 4.3, the vertices of the separating pairs can be put at the outer face of a K -component. Therefore the resulting graph is planar. However, this simple replacement we do only inside the tree, for K 5 -components K different from the root. Suppose path p from s to t reaches K at w 1 of separating pair {w 1,w 2 }. Then we only want to know whether there is a path from w 1 to w 2, because this is the only way to get back to the root node. Here it is fine to stick components together at their copies of a separating pair {w 1,w 2 }, because then we have the same paths available as in G. However, when the root of the tree, C i, is itself a K 5 -component, then the replacement is done differently. Consider the case that K is the root of T Ci, i.e., K = C i. Recall that C i has the separating pairs {a i 1,b i 1 } and {a i+1,b i+1 } as neighbors on path P from S to T. So now we have to consider four reachability questions from a i 1 or b i 1 to a i+1 or b i+1. For each of the four possibilities we create one CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

14 THOMAS THIERAUF AND FABIAN WAGNER copy of G i, say G i,1,...,g i,4. The corresponding trees T Ci,1,...,T Ci,4 have copies K 1,...,K 4 of K as their root. For example, at K 1 we ask for a path from a i to a i+1. Hence we identify w 1 of K 1 with a i and w 2 with a i+1. Then we replace each K i by a planar K i from Figure 4, adapted to the corresponding reachability problem. For the internal K 5 -nodes, we do the simple replacement described above. This yields subgraphs G i,1,...,g i,4. Graph G i is defined by identifying the copies of vertices a i 1,b i 1,a i+1,b i+1 in G i,1,...,g i,4, respectively. The drawing in Figure 5 shows that G i is planar. a i 1 G i,1 a i+1 G i G i,2 b i 1 G i,3 b i+1 G i,4 Figure 5: A schematic illustration of the construction when K is the root of T Ci. There are four reachability problems from the vertices of separating pairs {a i 1,b i 1 } to {a i+1,b i+1 }. For each of the four possibilities we have one copy of G i, say G i,1,...,g i,4. The corresponding trees T Ci,1,...,T Ci,4 have copies K 1,...,K 4 of K as their root. Then we replace each K i by a planar K i from Figure 4, adapted to the corresponding reachability problem. This yields subgraphs G i,1,...,g i,4. We join them at a i 1,b i 1,a i+1,b i+1 as indicated. This defines G i. Lemma 4.4. Let G i be the subgraph that results from G i after the replacement of the K 5 -components. G i has the following properties. (i) G i is planar. (ii) if C i is a component node: there are paths from a i 1 or b i 1 to a i+1 or b i+1 in G i if and only if there are such paths in G i. (iii) if C i is a separating pair node: there is a path from a i to b i in G i, or vice versa, if and only if there are such paths in G i. (iv) The size of G i is polynomial in the size of G i. Proof. By the discussion preceding the lemma it remains to show (i) and (iii). Ad (i). G i is constructed by merging planar components at their common separating pairs. Note that in each planar component the separating pairs are connected by a virtual edge. Therefore the two vertices of a separating pair touch the same face. Moreover, by Lemma 4.3, there is a planar embedding such that the root separating pair of each component is at the outer face. Therefore we can put all components with the same root separating pair inside a face which touches the root separating pair in the parent component. Hence the merging process yields again a planar graph. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

15 REACHABILITY IN K 3,3 -FREE AND K 5 -FREE GRAPHS IN UL Ad (iii). For G i of size N, let S(N) be the size of G i. If a K 5-component K in G i is replaced by a planar component K, some edges of K have several copies in K. If such an edge corresponds to a separating pair, we also copy the subtree of that edge. Let k be the number of additional copies of edges in K from K. We have the following recurrence for S(N), S(N) k S(N/2) + O(N). Recall that we do not make copies of a large child. The subgraphs we make copies of are of size N/2. This leads to a polynomial bound on S(N) for constant k : S(N) = O(N logk ). We give a bound on k : each of the four edges to w 5 in K have one extra copy in K, going to w 5,w 5, or w 5. Moreover, if K is at the root of the current subtree, we use the construction shown in Figure 5. Then we create four copies of the subtrees. Hence k 4 4 = 16. As a final step, we concatenate graphs G i at the separating pairs between them. The resulting graph G is the output of our reduction. Graph G is clearly planar and has polynomial size by Lemma 4.4 (iii). It is also biconnected. We argue that G can be constructed in logspace. Lemma 4.5. G is planar and can be computed in logspace. There is a path from s to t in G if and only if there is such a path in G. Proof. By Lemma 3.7 and 3.9 we can compute the triconnected component tree in logspace. We can also figure out which are the K 5 -components because they have constant size. We can compute path P from S to T in the tree in logspace. Recall that the tree is undirected. Then we replace the K 5 -components by the planar gadget. Thereby we distinguish whether a K 5 - component lies on path P or not and do the replacement accordingly as described above. From the resulting tree we construct G by identifying the copies of separating pairs. We mention some technical details of the algorithm. When a subtree is copied due to the replacement of a K 5 -component K by K, we give new names to vertices in the copies of the subtree. A separating pair in component K can have up to 8 copies in K (in case K is one of the root nodes). When the depth first traversal goes into recursion at a separating pair in K, we have to store at which copy of a separating pair we went into recursion. Because there are 8 copies, 3 bits suffice. We need such bits at each level of the recursion. In the depth traversal, whenever we reach a copied component for the first time, we relabel its vertices, keeping a counter on the work tape which starts by n + 1 (assuming, that G has vertices with labels 1,...,n). Then we recursively traverse the children of the component. At each stage in the tree, say at a node C, the sizes of the subtrees rooted at the children of C are 1/2 the size of T C. Hence, there are O(logn) levels of recursion and the algorithm runs in logspace. This finishes the proof of Theorem 4.1. Together with Lemma 3.5 and the result of [7] we get: Corollary 4.6. K 3,3 -free Reachability is in UL coul. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

16 THOMAS THIERAUF AND FABIAN WAGNER 4.3 Distance and longest paths in K 3,3 -free graphs For the distance problem and the longest path problem it suffices again to consider biconnected graphs, because we can pass only once through every articulation point on a simple path from s to t. Hence we can consider longest paths or distances in the biconnected components, and then sum up these lengths appropriately. For a biconnected K 3,3 -free graph G we use the same transformation to the planar graph G as above in Lemma 4.5. The crucial point to observe is that simple paths in a K 5 -component K of G have the same length as the corresponding paths in the planar component K in G. Hence, distances in G are the same as in G. Moreover, if G is acyclic, then also G is acyclic. Lemma K 3,3 -free Distance L T planar Distance. 2. K 3,3 -free DAG Long-Path L T planar DAG Long-Path. Thierauf and Wagner [24] proved that computing the distance in planar directed graphs is in UL coul. Limaye, Mahajan and Nimbhorkar [19] proved that computing a longest path in planar DAGs is in UL coul. Corollary 4.8. Distances in K 3,3 -free graphs and longest paths in K 3,3 -free DAGs can be computed in UL coul. 5 Reachability in K 5 -free graphs We give a logspace reduction from the reachability problem for directed K 5 -free graphs to the reachability problem for directed planar graphs. Theorem 5.1. biconnected K 5 -free Reachability L m planar Reachability. We consider again the decomposition of a biconnected graph into 3-connected components. The crucial theorem for K 5 -free graphs is due to Wagner [26]. Khuller [17] gives a formulation of Wagners theorem in terms of a clique-sum operation where graphs are joined at a common clique. Definition 5.2. Let G 1 = (V 1,E 1 ) and G 2 = (V 2,E 2 ) be two undirected graphs such that the induced subgraphs of G 1 and G 2 on V 1 V 2 both are cliques. A graph G = (V 1 V 2,E) is a clique-sum of G 1 and G 2 if E agrees with E 1 on V 1 V 2 and with E 2 on V 2 V 1, E is arbitrary on V 1 V 2, and there are no other edges in E. If V 1 V 2 k, we also say that G is a k-clique-sum. For a class G of graphs, G k is the closure of G under the k-clique-sum operation. The Möbius ladder M 8, is shown in Figure 6. It is a 3-connected graph on 8 vertices which is nonplanar, because it contains a K 3,3. Wagner showed that the closure under 3-clique-sum of planar graphs and the M 8 is precisely the class of K 5 -free graphs. Theorem 5.3. [26] Let C be the class of all planar graphs together with the Möbius ladder M 8. Then C 3 is the class of all graphs with no K 5 -minor. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

17 REACHABILITY IN K 3,3 -FREE AND K 5 -FREE GRAPHS IN UL Figure 6: The Möbius ladder M 8. We make two easy observations with respect to the above clique-sum operation. If we build the 3-clique-sum of two planar graphs, then the three vertices of the joint clique are a separating triple in the resulting graph. Hence the 4-connected components of a graph which is built as the 3-clique-sum of planar graphs must all be planar. The M 8 is nonplanar and 3-connected, but not 4-connected. Furthermore, the M 8 cannot be part of a 3-clique-sum operation where all the tree vertices are chosen from the M 8, because the M 8 does not contain a triangle as induced subgraph. By Theorem 5.3 and the two observations we get a characterization of all nonplanar 3-connected components of a K 5 -free graph. Corollary 5.4. (cf. [17]) A 3-connected nonplanar component of a K 5 -free biconnected graph is either the M 8 or its 4-connected components are all planar. In the next two sections we construct a planar gadget to replace the M 8 -components and show how to decompose the other nonplanar components into 4-connected planar components in logspace. 5.1 Transforming a M 8 -component into a planar component Recall again the partitioning of the reachability problem shown in Section 3.3. Let S = C 1, C 2,..., C l = T be the simple path from S to T in the triconnected component tree T of G. Let T Ci be the subtree rooted at C i and G i be the subgraph of G corresponding to T Ci. The C i s are alternating separating pair nodes, C i = {a i,b i } in this case, and component nodes. We traverse T Ci in depth first manner. When we reach an M 8 -component node, then we replace it locally by a planar component such that the reachability properties do not change. Let M be an M 8 -component node in T Ci with vertices w 1,w 2,...,w 8. If M is not the root of T Ci, then there is a parent separating pair, say {w 1,w 2 }, and we search for a path from w 1 to w 2. The planar graph M that replaces M is defined in Figure 7. Node M might have a large child within T Ci. The edge of a large child should not be copied. Figure 7 (b) and (c) show two cases where (w 3,w 4 ) and (w 1,w 3 ) correspond to a large child of M, respectively. If M is the root of T Ci, then we use the same construction as in Figure 5. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

18 THOMAS THIERAUF AND FABIAN WAGNER w 8 w 7 w 3 w 5. w 5 w 7 w 6 w 4 w 7 w 5. w 6 w 8 w 7 w 7 w 8w6 w 8 w w w5... w 7 w w 2 w 6 w w w w5 7 w 5 w 3 w w. 3 w. 5 w 8 w 6 w 6 w 4 w 8 w 1 w 7 w 6 w 4 w 2 w 7 w 8 w 1 w 3 w 8 w 6 w 5 w 4... w 6 w 4 w 4 w 2 w 7 w 7 w 8 (a) (b) (c) Figure 7: (a) Assume that we search for a path from w 1 to w 2 in the M 8 -component M. Edges are drawn undirected for simplicity. (b) The planar component M is shown schematically for the case that (w 3,w 4 ) corresponds to a large child of M. For simplicity, all the copies of a vertex in M have the same label in the picture. For every path from w 1 to w 2 that does not contain edge (w 3,w 4 ), M contains a copy of the path, i.e. a copy of all vertices and edges on this path. This is indicated by the paths above the edge (w 1,w 2 ) in the picture. The remaining paths go along (w 3,w 4 ) or (w 4,w 3 ) in M. This edge should occur only once in M. Therefore, these paths in M are subdivided into paths from w 1 to w 3 or to w 4, and from these vertices to w 2. This is indicated in the part below the edge (w 1,w 2 ) in the picture. (c) M in the case that (w 1,w 3 ) is the large child. The construction is essentially the same as in (b). Lemma 5.5. Let M be a M 8 -component node with vertices w 1,w 2,...,w 8, where we search for a path from w 1 to w 2, and, if there is a large child, it is at (w 3,w 4 ). Let M be the component constructed from M shown in Figure 7 (b). Component M has the following properties: M is planar. Every path from w 1 to w 2 in M exists as well in M, possible going through the copies of the vertices instead. M contains the edge (w 3,w 4 ) only once. Vertices w 1 and w 2 are both on the outer-face of M in the embedding shown in Figure 7. The second item of the lemma implies the correctness of the construction: There is a path from w 1 to w 2 in M or vice versa, iff there is such a path in M. We put things together. Analogously to Lemma 4.4 and 4.5, we get: CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages

19 REACHABILITY IN K 3,3 -FREE AND K 5 -FREE GRAPHS IN UL Lemma 5.6. Let G i be the subgraph that results from G i after the replacement of the M 8 -components. G i has the following properties: (i) G i has no M 8 as a 3-connected component. (ii) if C i is a component node: there are paths from a i 1 or b i 1 to a i+1 or b i+1 in G i if and only if there are such paths in G i. (iii) if C i is a separating pair node: there is a path from a i to b i in G i, or vice versa, if and only if there are such paths in G i. (iv) The size of G i is polynomial in the size of G i. (v) G i can be constructed in logspace. The proof is the same as for the corresponding lemmas in Section 4. The constant k in the size bound for G i is a bit larger here, because we have more copies of vertices. The construction algorithm works also in the same way, we just have to replace a M 8 -component here instead of a K 5 -component. 5.2 The 4-connected component tree We show how to further decompose the 3-connected components which are nonplanar and not the M 8. We start by identifying the separating triples that define a 4-connected component. For a given separating triple τ 0, we would like to define the maximal separating triples w.r.t. τ 0 as the separating triples τ that are not separated from τ 0 by any other separating triple. These separating triples define one 4-connected component. A technical difficulty here is that the split components of two separating triples might overlap each other. If we would simply split a 3-connected component along all its separating triples, we might decompose some parts more than once. To avoid this, we first define the notion of a candidate maximal separating triples. The actual maximal separating triples to proceed with are then selected from the candidates. Definition 5.7. Let C be a 3-connected component and τ 0 be a separating triple in C. Let C be a split component of τ 0 in C. A separating triple τ τ 0 is a candidate maximal separating triple in C w.r.t. τ 0, if no separating triple τ {τ,τ 0 } separates τ from τ 0 in C. Two candidate maximal separating triples τ 1,τ 2 w.r.t. τ 0 are called crossing, if there is a vertex v τ 0 τ 1 τ 2 in C which is separated from τ 0 by τ 1 and by τ 2. Figure 8 shows an example where crossing separating triples occur. The split component of separating triple τ 0 = {a 0,b 0,c 0 } in Figure 8 (a) has separating triples τ 1 = {a,b 0,c 0 } and τ 2 = {a 0,b 0,c} which overlap each other and τ 0. Figure 8 (b) and (c) show the split components we get when we split further along τ 1 and τ 2. The split components overlap each other in vertex b. Therefore τ 1 and τ 2 are crossing. The next lemma shows that there are essentially two ways how crossing candidate maximal separating triples can occur. Figure 8 and 11 present the two possibilities. Lemma 5.8. Let τ 1 and τ 2 be two crossing candidate maximal separating triples w.r.t. τ 0 in a triconnected component C. Then τ 0 τ 1 τ 2 and τ 1 τ 2 1. Furthermore, τ 1 and τ 2 each have exactly one split component besides the one with τ 0. CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 2, pages